Question-20
Let \(X=aZ+b\), where \(Z\) is a standard normal random variable, and \(a,b\) are two unknown constants. It is given that
\[\begin{gather*} E[ X] =1,\ \ E[( X-E[ X]) Z] =-2,\ \ \\ \\ E\left[( X-E[ X])^{2}\right] =4 \end{gather*}\]
where \(E[ X]\) denotes the expectation of the random variable \(X\). The values of \(a,b\) are:
Since \(Z\) is a standard normal random variable, \(E[ Z] =0\) and \(E\left[ Z^{2}\right] =1\). Also, given \(X=aZ+b\), we have \(E[ X] =b\) and \(E\left[( X-E[ X])^{2}\right] =a^{2}\). From the given data, we see that \(b=1\) and \(a=\pm 2\). Using the final piece of information:
\[ \begin{array}{ r l c } & E[( X-E[ X]) Z] & =-2\\ \Longrightarrow & E\left[ aZ^{2}\right] & =-2\\ \Longrightarrow & aE\left[ Z^{2}\right] & =-2\\ \Longrightarrow & a & =-2 \end{array} \]